Bayes risk and minimax risk

Let $(\mathrm{\Theta },\mathcal{T})$ and $(\mathcal{X},\mathcal{A})$ be measurable parameter and data spaces, with $\mathrm{\Theta }$ nonempty, and let $K:\mathrm{\Theta }\times \mathcal{A}\to [0,1]$ be a probability kernel, writing $P_{\theta }=K(\theta ,\cdot )$ for the sampling law (Parameter-measurable statistical models as kernels). Let $(\mathsf{D},\mathcal{G})$ be a nonempty measurable action space (Measurable spaces and measurable sets). Let $L:\mathrm{\Theta }\times \mathsf{D}\to \bar{\mathbb{R}}$ be measurable from the product measurable space with sigma-algebra $\mathcal{T}\otimes \mathcal{G}$ to the extended real line with its Borel sigma-algebra, and assume $L(\theta ,a)\ge 0$ for all $(\theta ,a)$ (The product sigma-algebra and its finite iterates, A measurable function between measurable spaces, The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined, The Borel sigma-algebra on the extended real line). A prior $\pi$ is a probability measure on $\mathrm{\Theta }$. The risk function $\theta \mapsto R(\theta ,\delta )$ is measurable under these hypotheses by kernel integration (Parameter-measurable statistical models as kernels, Measurability of integration against a kernel). The Bayes risk of $\delta$ under $\pi$ is $r_{\pi }(\delta )={\int }_{\mathrm{\Theta }}R(\theta ,\delta )\pi (d\theta )\in [0,+\mathrm{\infty }]\mathrm{.}$ Its worst-case risk is $R^{\ast }(\delta )={\sup }_{\theta \in \mathrm{\Theta }}R(\theta ,\delta )\in [0,+\mathrm{\infty }],$ and the minimax value is $V_{\mathrm{m}\mathrm{m}}=\inf \{R^{\ast }(\delta ):\delta \text{ is a measurable randomized rule from data to actions}\}\in [0,+\mathrm{\infty }]\mathrm{.}$ These suprema and infima are taken in the extended order on $\bar{\mathbb{R}}$ (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined); they exist by Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$. The parameter set is nonempty and the rule class is nonempty under the action-space hypothesis, so these are suprema and infima of nonempty sets of nonnegative extended risks. A rule is minimax if its worst-case risk equals this infimum. These definitions make no assertion that a minimizer exists. Nonemptiness of $\mathsf{D}$ ensures the rule class is nonempty: fixing any $a_0\in \mathsf{D}$ gives the constant point-mass rule $x\mapsto {\delta }_{a_0}$.